GARCH and Volatility Clustering

Modeling the autocorrelation of squared returns.

Asset returns aren't autocorrelated, but their squared values are: big moves cluster with big moves, calm with calm. GARCH $(p, q)$ models this volatility clustering:

\sigma_t^2 = \omega + \sum_{i=1}^{p} \alpha_i \epsilon_{t-i}^2 + \sum_{j=1}^{q} \beta_j \sigma_{t-j}^2

GARCH(1,1), the simplest workhorse, captures most of the empirical volatility behavior. The coefficient $\alpha + \beta$ controls persistence — close to $1$ means shocks linger.

GARCH is the standard volatility model for risk management, options pricing in the absence of options data, and Value-at-Risk calculation. Extensions handle asymmetry (EGARCH, TGARCH for the leverage effect: drops increase volatility more than rallies of equal size) and longer memory (FIGARCH).

Limit: GARCH still implicitly assumes Gaussian-conditional returns. Real residuals are leptokurtic; using Student-t innovations or hybrid models often improves tail fit.